Question

limites, alguém poderia me ajudar

Answer 1

Olá.

$\displaystyle f(x) = \frac{1}{x} \\ \\ \\ f(p) = \frac{1}{p}$

Portanto:

$\displaystyle \lim_{x \to p} \frac{f(x)-f(p)}{x-p} \\ \\ \\ \lim_{x \to p} \frac{\displaystyle \frac{1}{x}-\frac{1}{p}}{x-p} \\ \\ \\ \lim_{x \to p} \frac{\displaystyle \frac{p-x}{xp}}{x-p} \\ \\ \\ \lim_{x \to p} \frac{p-x}{xp} \cdot \frac{1}{x-p} \\ \\ \\ \lim_{x \to p} \frac{p-x}{xp \cdot (x-p)} \\ \\ \\ \lim_{x \to p} \frac{(p-x)}{-xp \cdot (p-x)} \\ \\ \\ \lim_{x \to p} -\frac{1}{xp} \\ \\ \\ \lim_{x \to p} -\frac{1}{p \cdot p} = \boxed{\boxed{-\frac{1}{p^2}}}$